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Theorem toponcom 21633
 Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
toponcom ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐾))

Proof of Theorem toponcom
StepHypRef Expression
1 toponuni 21619 . . . 4 (𝐾 ∈ (TopOn‘ 𝐽) → 𝐽 = 𝐾)
21eqcomd 2764 . . 3 (𝐾 ∈ (TopOn‘ 𝐽) → 𝐾 = 𝐽)
32anim2i 619 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → (𝐽 ∈ Top ∧ 𝐾 = 𝐽))
4 istopon 21617 . 2 (𝐽 ∈ (TopOn‘ 𝐾) ↔ (𝐽 ∈ Top ∧ 𝐾 = 𝐽))
53, 4sylibr 237 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∪ cuni 4801  ‘cfv 6339  Topctop 21598  TopOnctopon 21615 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6298  df-fun 6341  df-fv 6347  df-topon 21616 This theorem is referenced by:  toponcomb  21634  kgencn3  22263
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